integrate sinx/sin4x

∫(sinx)/(sin4x) dx

=∫1/4cosxcos2x  dx

=1/4 ∫cosx dx /cos²xcos2x

==1/4 ∫cosx dx /(1-sin²x)(1-2sin²x)

let sinx =t

   cosx dx =dt

=1/4 ∫dt /(1-t²)(1-2t²)

use partial fraction

=-1/4 ∫dt /(1-t²)   + 1/8 ∫dt /(1-2t²)

now onwards it is easily calculated